Let's say we have two positive semi-definite matrices $A$ and $B$ and a negative real $\lambda$. What would be the conditions for the matrix $M$, defined as follows, to be positive semi-definite too ? $$M = \begin{pmatrix} A & \lambda I\\ \lambda I & B \end{pmatrix},$$ where $I$ is the identity matrix.
I know that $M$ is positive semi-definite if and only if its Schur complement $S=A- \lambda^2 B$ is positive semi-definite, as it is stated here. But it doesn't help me much... Furthermore, I would like to find conditions which work also for block matrices with more than 2 blocks per row and columns like this one: $$\begin{pmatrix} A & \lambda_{1,2} I & \lambda_{1,3} I \\ \lambda_{1,2} I & B & \lambda_{2,3} I \\ \lambda_{1,3} I & \lambda_{2,3} I & C \end{pmatrix},$$